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2018 | Buch

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Calculus of Variations

verfasst von: Dr. Filip Rindler

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook provides a comprehensive introduction to the classical and modern calculus of variations, serving as a useful reference to advanced undergraduate and graduate students as well as researchers in the field.

Starting from ten motivational examples, the book begins with the most important aspects of the classical theory, including the Direct Method, the Euler-Lagrange equation, Lagrange multipliers, Noether’s Theorem and some regularity theory. Based on the efficient Young measure approach, the author then discusses the vectorial theory of integral functionals, including quasiconvexity, polyconvexity, and relaxation. In the second part, more recent material such as rigidity in differential inclusions, microstructure, convex integration, singularities in measures, functionals defined on functions of bounded variation (BV), and Γ-convergence for phase transitions and homogenization are explored.

While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study. The reader is assumed to be familiar with basic vector analysis, functional analysis, Sobolev spaces, and measure theory, though most of the preliminaries are also recalled in the appendix.

Inhaltsverzeichnis

Frontmatter

Basic Course

Frontmatter

Chapter 1. Introduction

Abstract

In the quest to formulate useful mathematical models of aspects of the world, it turns out on surprisingly many occasions that the model becomes clearer, more compact, or more tractable if one introduces some form of variational principle. This means that one can find a quantity, such as energy or entropy, which obeys a minimization, maximization or saddle-point law.

Filip Rindler

Chapter 2. Convexity

Abstract

In this chapter we start to develop the mathematical theory that will allow us to analyze the problems presented in the introduction, and many more. The basic minimization problem that we are considering is the following:

$$ \left\{ \begin{aligned}&\text {Minimize} \quad \mathscr {F}[u] := \int _\varOmega f(x, u(x), \nabla u(x)) \,\mathrm{d}x\\&\text {over all} \quad \quad u \in \mathrm {W}^{1,p}(\varOmega ;\mathbb {R}^m)\text { with }u|_{\partial \varOmega } = g. \end{aligned} \right. $$

Filip Rindler

Chapter 3. Variations

Abstract

In this chapter we discuss variations of functionals. The idea is the following: Let $\mathscr {F}:\mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m) \rightarrow \mathbb {R}$ be a functional with minimizer $u_* \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)$. Take a path $t \mapsto u_t \in \mathrm {W}^{1,p}_g(\varOmega ;\mathbb {R}^m)$ ($t \in \mathbb {R}$) with $u_0 = u_*$ and consider the behavior of the map

$$ t \mapsto \mathscr {F}[u_t] $$

around $t = 0$. If $t \mapsto \mathscr {F}[u_t]$ is differentiable at $t = 0$, then its derivative at $t = 0$ must vanish because of the minimization property. This is analogous to the elementary fact that if $g \in \mathrm {C}^1((0,T))$ takes its minimum at a point $t_* \in (0,T)$, then $g'(t_*) = 0$.

Filip Rindler

Chapter 4. Young Measures

Abstract

Before we continue our study of integral functionals, we first introduce an abstract, yet very versatile, tool, the Young measure, named after its inventor Laurence C. Young. Young measures pervade much of the modern theory of the calculus of variations and will be used throughout the remainder of the book. In the next chapter, we will see their first use in the proof of the lower semicontinuity theorem for integral functionals with quasiconvex integrands.

Filip Rindler

Chapter 5. Quasiconvexity

Abstract

We saw in the Tonelli–Serrin Theorem 2.6 that convexity of the integrand (in the gradient variable) implies the weak lower semicontinuity of the corresponding integral functional. Moreover, we proved in Proposition 2.9 that if $d = 1$ or $m = 1$, then convexity of the integrand is also necessary for weak lower semicontinuity. In the vectorial case ($d, m > 1$), however, it turns out that one can find weakly lower semicontinuous integral functionals whose integrands are non-convex.

Filip Rindler

Chapter 6. Polyconvexity

Abstract

At the beginning of the previous chapter we saw that convexity cannot hold concurrently with frame-indifference (and a mild non-degeneracy condition). Thus, we were led to consider quasiconvex integrands. However, while quasiconvexity is of tremendous importance in the theory of the calculus of variations, our Lower Semicontinuity Theorem 5.16 has one major drawback: we needed to require the p-growth bound

$$ |f(x, A)| \le M(1+|A|^p), \qquad (x, A) \in \varOmega \times \mathbb {R}^{m \times d}, $$

Filip Rindler

Chapter 7. Relaxation

Abstract

Consider the functional

$$ \mathscr {F}[u] := \int _0^1 |u(x)|^2 + \bigl (|u'(x)|^2-1 \bigr )^2 \;\mathrm{d}x, \qquad u \in \mathrm {W}^{1,4}_0(0,1). $$

The gradient part of the integrand, $a \mapsto (a^2-1)^2$, see Figure 7.1, has two distinct minima, which makes it a double-well potential. Approximate minimizers of $\mathscr {F}$ try to satisfy $u' \in \{-1, 1\}$ as closely as possible, while at the same time staying close to zero because of the first term.

Filip Rindler

Advanced Topics

Frontmatter

Chapter 8. Rigidity

Abstract

We noted in several places that oscillations may develop in minimizing sequences. Now we will embark on a more detailed study of these oscillations. Inspired by (but not limited to) the example on crystalline microstructure in Section 1.8, our overarching philosophy is the following: Assume that we are trying to minimize the functional

$$ \mathscr {F}[u] := \int _\varOmega f(\nabla u(x)) \;\mathrm{d}x,$$

where $f :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ ($d, m \ge 2$) is continuous, over a (Sobolev) class of functions $u :\varOmega \rightarrow \mathbb {R}^m$ with prescribed boundary values. Here, as usual, we assume that $\varOmega \subset \mathbb {R}^d$ is a bounded Lipschitz domain. We associate with $\mathscr {F}$ as above the pointwise differential inclusion

$$ \nabla u(x) \in K := \bigl \{\, A \in \mathbb {R}^{m \times d} \ \ \mathbf : \ \ f(A) = \min f \,\bigr \}, \qquad x \in \varOmega ,$$

where $\min f$ denotes the pointwise minimum of f that we assume to exist in $\mathbb {R}$. Under a mild coercivity assumption on f we have that K is compact.

Filip Rindler

Chapter 9. Microstructure

Abstract

Motivated by the example on crystal microstructure in Section 1.8 and the remarks in Section 8.3 about the connection of the quasiconvex hull to the relaxation of integral functionals, in this chapter we continue our analysis of the differential inclusion

$$\begin{aligned} \left\{ \begin{aligned}&u \in \mathrm {W}^{1,\infty }(\varOmega ;\mathbb {R}^m), \quad u|_{\partial \varOmega } = Fx,\\&\nabla u \in K \quad \text {in}\, \varOmega . \end{aligned}\right. \end{aligned}$$

Filip Rindler

Chapter 10. Singularities

Abstract

All of the existence theorems for minimizers of integral functionals defined on Sobolev spaces $\mathrm {W}^{1,p}(\varOmega ;\mathbb {R}^m)$ that we have seen so far required that $p > 1$. Extending the existence theory to the linear-growth case $p=1$ turns out to be quite intricate and necessitates the development of new tools.

Filip Rindler

Chapter 11. Linear-Growth Functionals

Abstract

After the preparations in the previous chapter, we now return to the task at hand, namely to analyze the following minimization problem for an integral functional with linear growth:

$$ \left\{ \begin{aligned}&\text {Minimize} \quad {\mathscr {F}}[u] := \int _\varOmega f(x, \nabla u(x)) \;\text {d}x\\&\text {over all} \quad \quad u \in \mathrm {W}^{1,1}(\varOmega ;{\mathbb {R}}^m) \,\text {with}\, u|_{\partial \varOmega } = g. \end{aligned} \right. $$

Filip Rindler

Chapter 12. Generalized Young Measures

Abstract

In this chapter we continue the study of the integral functional

$$ \mathscr {F}[u] := \int _\varOmega f(x, \nabla u(x)) \;\mathrm{d}x + \int _\varOmega f^\# \biggl ( x, \frac{\mathrm{d}D^s u}{\mathrm{d}|D^s u|}(x) \biggr ), \qquad u \in \mathrm {BV}(\varOmega ;\mathbb {R}^m), $$

for a Carathéodory integrand $f :\varOmega \times \mathbb {R}^{m \times d} \rightarrow \mathbb {R}$ with linear growth. In contrast to the preceding chapter, however, here we proceed in a more abstract way: We first introduce the theory of generalized Young measures, which extends the standard theory of Young measures developed in Chapter 4. Besides quantifying oscillations (like classical Young measures), this theory crucially allows one to quantify concentrations as well, thus providing a rich toolbox for investigating linear-growth functionals. While the (generalized) Young measure approach requires a fair bit of abstract theory, the initial effort is rewarded with a robust general framework that has become a core tool in the calculus of variations with applications way beyond the lower semicontinuity theory of integral functionals.

Filip Rindler

Chapter 13. -Convergence

Abstract

Often, a functional of interest depends on the value of a parameter, say a small $\varepsilon > 0$, as we have seen with the functionals $\mathscr {F}_\varepsilon $ from the examples on phase transitions and composite elastic materials in Sections 1.9 and 1.10, respectively. In these cases the goal often lies not in minimizing $\mathscr {F}_\varepsilon $ for one particular value of $\varepsilon $, but in determining the asymptotic limit of the minimization problems as $\varepsilon \downarrow 0$. Concretely, we need to identify, if possible, a limit functional $\mathscr {F}_0$ such that the minimizers and minimum values of the $\mathscr {F}_\varepsilon $ (if they exist) converge to the minimizers and minimum values of $\mathscr {F}_0$ as $\varepsilon \downarrow 0$.

Filip Rindler

Backmatter

Titel: Calculus of Variations
verfasst von: Dr. Filip Rindler
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-77637-8
Print ISBN: 978-3-319-77636-1
DOI: https://doi.org/10.1007/978-3-319-77637-8